Abelian returns in Sturmian words

In this paper we study an abelian version of the notion of return word. Our main result is a new characterization of Sturmian words via abelian returns. Namely, we prove that a word is Sturmian if and only if each of its factors has two or three abelian returns. In addition, we describe the structure of abelian returns in Sturmian words, and discuss connections between abelian returns and periodicity.

💡 Research Summary

The paper introduces an abelian version of the classical notion of return words and uses it to obtain a new characterization of Sturmian words. A return word of a factor v in an infinite recurrent word w is traditionally defined as the factor that starts at an occurrence of v and ends just before the next occurrence of v. The authors replace exact equality by abelian equivalence (two words are abelian‑equivalent if they contain the same number of each letter), thereby defining “abelian returns”: for each occurrence of a factor whose abelian class matches v, the segment up to the next occurrence of a word from the same abelian class is taken, and the set of such segments is considered up to abelian equivalence.

The main theorem (Theorem 2) states that an aperiodic recurrent infinite word is Sturmian if and only if every factor has exactly two or three abelian returns. This mirrors the classical result that a word is Sturmian precisely when each factor has two ordinary returns, but the abelian setting allows a third return in certain cases.

The authors first establish a simple sufficient condition for periodicity (Lemma 1): on an alphabet of size k, if every factor has at most k abelian returns, the word must be periodic. The proof uses the fact that a factor containing all letters forces a full set of distinct abelian returns, which forces a fixed period. They note that this condition is not necessary, providing examples of periodic words with more than k abelian returns.

To prove the necessity part of the main theorem, the paper analyses the fine structure of Sturmian words. It recalls standard notions such as right‑special, left‑special, bispecial, and singular factors. In Sturmian words there is exactly one right‑special factor of each length, and singular factors are those whose abelian class contains only themselves; they have the form a B a where a is a letter and B is bispecial. Proposition 2 shows that any abelian return of a Sturmian factor is either a single letter or a word of the form a B b with a ≠ b and B bispecial. Corollary 1 further limits the number of non‑trivial (length ≥ 2) abelian returns to at most one per length.

Using these combinatorial constraints, Proposition 3 proves that every Sturmian factor indeed has only two or three abelian returns. The proof proceeds by contradiction: assuming a factor has four distinct abelian returns leads to an impossible configuration in the lexicographic array (a matrix that records all cyclic shifts of a balanced word). The argument exploits the balanced property of Sturmian words (the number of 1’s in any two factors of the same length differs by at most one) and the ordering of the cyclic shifts.

The sufficiency direction (Section 6) shows that if an aperiodic word has the property that each factor possesses at most three abelian returns, then the word must be Sturmian. The authors use Lemma 1 to rule out periodicity and then argue that the only way to keep the number of abelian returns bounded by three is to satisfy the balanced condition with exactly one right‑special factor per length, which characterizes Sturmian words. Theorem 4 refines the picture by proving that a factor has exactly two abelian returns if and only if it is singular.

A technical tool employed throughout the paper is the lexicographic array of a balanced word. For a binary word of length q with p occurrences of 1 (gcd(p,q)=1), the q cyclic shifts are arranged in lexicographic order, forming a q × q matrix. The array makes it possible to read off the positions and lengths of abelian returns directly: columns correspond to possible return lengths, and rows to successive occurrences of the factor’s abelian class. The authors illustrate the method with concrete examples (e.g., the abelian class of 001) and show how the array guarantees the claimed bounds on the number of returns.

In summary, the paper extends the classical return‑word framework to the abelian setting, proves that the “two‑or‑three abelian returns per factor” condition precisely captures Sturmian words, and provides a detailed combinatorial description of the possible abelian returns. The results deepen the connection between abelian combinatorics, balanced words, and symbolic dynamics, and open the way for further investigations of abelian return structures in other low‑complexity word families.